A fundamental driving question in algebraic geometry is how positivity of the canonical controls the geometric, analytic and arithmetic properties of varieties. We describe two big conjectures in this vein, the Lang Conjecture and the Kobayashi Conjecture, and talk about the relation between them. Discussing the difference between the results of these two conjectures naturally leads to the question of which varieties should be hyperbolic. We discuss the notion of algebraic hyperbolicity as an easier-to-check algebraic property that is conjecturally equivalent to being hyperbolic and describe some recent work proving algebraic hyperbolicity for particular surfaces of general type. Some of this is joint work with David Yang and some of this is joint with Izzet Coskun.

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